 \documentclass[12pt]{article}
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\usepackage{hyperref}
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\usepackage{url}
\usepackage{pgfplots}
\usepackage{subcaption}
\usepackage{epstopdf}
\graphicspath{{./Graphs/}}
\usepackage{chngcntr}
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\newtheorem{definition}{Definition}
\newtheorem{claim}{Claim}
\newtheorem{example}{Example}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{corollary}{Corollary}
\newtheorem{implication}{Implication}
\newtheorem{assumption}{Assumption}
\newtheorem{property}{Property}
\newcommand{\bit}{\begin{itemize}}


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\usepackage {color}
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\numberwithin{property}{section}
\usepackage{graphicx}
\usepackage{tikz}
\graphicspath{ {./Graphs/} }
\begin {document} 


\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
    axis lines = middle,
    xlabel = $x_1$,
    ylabel = {$E[U_L(x_1)],
     E[U_V(x_1)]$}, 
    y label style={at={(axis description cs:0.5,-0.1)},anchor=north},
    x label style={at={(axis description cs:1,1)},anchor=south},
    yticklabels={,,,,},
    xticklabels={,,,}
]
  \addplot[color=black,solid,thick,mark=*]
   coordinates {
       
        (1/3,0)
      
        };
        
     
        \node[below]  at (axis cs: 1/3,-0.1) {$\overline{\alpha}(2\gamma_V-1)$};
        
    
\addplot [
    domain=-1:-1/2, 
    samples=100, 
    color=black,
    style=thick,]
{-(x+1)^2};
\addplot [
    domain=-1/2:0, 
    samples=100, 
    color=black,
    style=thick,
]
{-(x+1)^2-(1+2*x)*(4/3)^2};

\addplot [
    domain=1/2:1.1, 
    samples=100, 
    color=black,
    style=thick,
]
{-(x+1)^2};

\addplot [
    domain=0:1/2, 
    samples=100, 
    color=black,
    style=thick,
]
{-(x+1)^2-(1-2*x)*(4/3)^2};

\addplot[
domain=-1:1,
samples=100,
 color=black,
 style=dashed,
]
{-2/3*(x-1)^2-1/3*(x+1)^2};

\addplot[color=blue,dashed,thick, mark=*]
   coordinates {
         
              (1/2,0)
            (1/6,0)

};
\node[below]  at (axis cs: 1/2,-1.7) {$x^{Pos}_L$};
\end{axis} 
\end{tikzpicture}
\caption{Players' utility as a function of first-period policy. The solid line represents the left-wing party's expected utility in the whole game, while the dashed line represents the voter's first-period expected utility.}
\end{center}
\end{figure}
\end{document} 